User blog:Cloudy176/Simulating PEGG
A day or two ago, PsiCubed2 created an ever-growing googolism, or PEGG for short. I think this is a great idea, and I wanted to try out the idea a bit further, and wanted to know what will happen to PEGG in the future. So I decided to simulate it. Since I don't think I can predict future values of Dow Jones, I slightly modified the definition of PEGG for our simulation purposes. Specifically, rule (a) in the definition is replaced with the following: (a) We take the (days passed since 2017-05-08)-th integer in the result from the Random Integer Generator by random.org, where the minimum value is 1, the maximum value is 10, and the pregenerated randomization from 2017-05-08 is used (you'll need to switch to Advanced Mode to set this). Call it n. The first 100 integers produced using the generator are: 10 2 8 8 3 2 9 4 9 5 7 3 8 4 4 8 5 9 3 9 8 2 8 7 2 5 5 4 2 8 2 4 6 4 5 3 4 3 1 5 2 3 5 10 3 7 10 1 8 1 8 3 2 4 2 3 2 10 9 1 7 2 10 2 6 6 6 1 6 6 5 10 5 1 4 8 10 3 2 8 4 2 3 6 3 9 5 7 4 6 6 1 1 6 2 10 6 9 3 9 (The numbers are read from left-to-right, then downwards.) Using these values, I simulated what happens in the first 100 days in this modified definition of PEGG. At the beginning, Y = "E", Z = 0.01, and K = 1000. The simulation starts as follows (See this post for the defintion of Letter Notation): At this point, we should switch to the letter F, and need to find a value m such that Fm = E10.43. Since rule (e-3) only requires us to reach a certain digits of precision after decimal point (currently 3 digits), we can use, say, binary search to determine the value. Some calculations later, I found: * F2.007 = EE(10^0.007) = EE1.0162... = E10.3812... * F2.008 = EE(10^0.008) = EE1.0185... = E10.4373... Therefore F2.007 < E10.43 < F2.008. So the new value for X is 2.007. (Edit: Actually, you don't need binary search. Just reverse the calculation: E10.43 = EE(log(10.43)) = EE1.01828... = F(2+log(1.01828...)) = F2.00786...) The values are updated to Y = "F", Z = 0.007, and K = 10000, and the simulation continues: (In the table below, (10↑)A B means 10↑10↑...10↑B with A repetitions of "10↑") (Notice that the simulated PEGG first surpasses a googol on June 4, and a googolplex on June 10.) At this point, another switch is required. Since 10.001 is very close to 10, we expect that the new value of X is 2. In fact: * G2.0000 = FF1 = F10 * G2.0001 = FF(10^0.0001) = FF1.00023... = FE(10^0.00023...) = FE1.00053... = F10.01222... Therefore G2.0000 < F10.001 < G2.0001, and the new value for X is 2. (Edit: Using the reverse-calculation method, F10.001 = FE1.000043... = FF1.000018... = G2.0000081...) The values are updated to Y = "G", Z = 0.0049, and K = 100000, and the simulation continues: (In the table below, (10↑↑)A B means 10↑↑10↑↑...10↑↑B with A repetitions of "10↑↑") This concludes the first 100 days of the simulation of (modified) PEGG. I think the real PEGG would behave similarly, provided that the "Dow Johns rule" indeed produces pseudorandom numbers. Category:Blog posts